If X X is a graph, κ \kappa a cardinal, then there is a graph Y Y such that if the vertex set of Y Y is κ \kappa -colored, then there exists a monocolored induced copy of X X ; moreover, if X X does not contain a complete graph on α \alpha vertices, neither does Y Y . This may not be true, if we exclude noncomplete graphs as subgraphs. It is consistent that there exists a graph X X such that for every graph Y Y there is a two-coloring of the edges of Y Y such that there is no monocolored induced copy of X X . Similarly, a triangle-free X X may exist such that every Y Y must contain an infinite complete graph, assuming that coloring Y Y ’s edges with countably many colors a monocolored copy of X X always exists.